J. S. Geronimo, D. P. Hardin and P. R. Massopust [J. Approximation Theory 78, No. 3, 373–401 (1994; Zbl 0806.41016)] first extended a piecewise polynomial scaling function, the linear B–spline, to an orthogonal scaling vector (commonly referred to as the GHM scaling vector), by adding a function supported on \([0, 1]\) (hence, automatically supported by its integer translates) that, when its integer translates were projected out of the original function, made the resulting function orthogonal to its integer translates. Since then, this basic idea has been applied by several authors for various purposes. All of these applications exploit the general strengths of using multiwavelets, namely the ability to build symmetric scaling functions of relatively short support, but they also suffered from the general weakness of multiwavelets, namely that the filters associated with a general scaling function of aproximation order \(K\) do not necessarily preserve discrete discrete–time polynomial data of degree \(K-1\). One possible way of dealing with this shortcoming is to pefilter the raw data. A more recent approach, initiated by C. Lebrun and M. Vetterli in [IEEE Trans. Signal Process. 46, (4), 1119–1125 (1998)] and [Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), vol. 3, Seattle, May 1998, 1529–1532] and studied further by C. K. Chui and Q. Jiang in [Math. Comp. 74, 1323–1344 (2005; Zbl 1061.42023)] and [Modern developments in multivariate approximation. Proceedings of the 5th international conference, Witten-Bommerholz, Germany, September 22–27, 2002. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 145, 71-102 (2003 (2003; Zbl 1034.42035)], the author [Approximation Theory XI, Gatlinburg, 2004, Nashboro Press, Brentwood, TN, 2005, 197–208 (2005)] and others, is to design scaling vecors whose filters maintain polynomial order without prefiltering, called balanced multiwavelets.
The purpose of the paper under review is to determine whether the useful trick of adding functions with support on \([0, 1]\) can be used to extend the spline–based scaling vectors to scaling vectors that are both orthogonal and balanced up to their approximation order. The macroelement approach is natural, since the functions considered are either suported completely on \([0, 1]\) or piecewise polynomial with integer knots. Following a brief introduction on notation and terminology in Section 1, the author shows in Section 2 that the two conditions cannot be met simultaneously using this type of construction. However, this technique can be used to design dual biorthogonal scaling vectors where the analysis basis is balanced. Two general constructions of scaling vectors with symmetry properties and approximation orders two and three, respectively, are shown in Section 3, with two concrete examples of each construction provided. The coefficient matrices satisfying the dilation equations for these scaling vectors are given in an appendix.